volterra integral equations

This work addresses the challenges of opaque memory mechanisms and difficulties in long-horizon training inherent in non-Markovian time series modeling by proposing Volterra signatures as an explicit representation of historical dependencies. These signatures are derived by expanding input paths with temporal kernels within a tensor algebra framework, and theoretical learning guarantees are established via the Volterra–Chen identity. Under augmented conditions, we prove the identifiability of these signatures and their universal approximation capability over infinite-dimensional path spaces—requiring only linear functionals in certain cases. Moreover, we show that their inner product admits a closed-form expression through a two-parameter integral equation, enabling compatibility with kernel methods and numerical PDE solvers. Experiments demonstrate that the proposed approach significantly outperforms classical path signature baselines on both real-world and synthetic dynamical tasks, achieving superior expressiveness while maintaining computational feasibility.

Efficient simulation of path-dependent Volterra-type stochastic differential equations (non-Markovian) remains challenging due to their intrinsic infinite memory. Method: This paper proposes a reversible embedding framework based on convolutional kernels that equivalently transforms the original process into a standard Markovian diffusion, modeling long-range dependence via a “goldfish memory” mechanism. Contributions/Results: Theoretically, we establish existence and Hölder regularity of solutions. Methodologically, we achieve the first rigorous, invertible non-Markov-to-Markov transformation and design a novel numerical scheme whose strong convergence order is uniformly 1/2—robust with respect to the roughness parameter and breaking the accuracy ceiling of Euler-type methods. Validated in volatility modeling, theoretical guarantees are confirmed numerically: all experiments consistently exhibit the predicted 1/2 strong convergence rate, significantly outperforming conventional approaches.

This paper addresses nonlinear filters in Euclidean space that are causal, time-invariant, and possess fading memory. We propose a universal Volterra reservoir kernel. Methodologically, we first integrate the state-space representation of Volterra series with reservoir computing, constructing a recursively computable kernel mapping on an infinite-dimensional tensor algebra—balancing theoretical universality and practical implementability. Theoretically, the kernel satisfies the Representer Theorem and uniformly approximates any fading-memory filter. Empirically, it significantly improves modeling accuracy and generalization in Bitcoin price forecasting—a challenging nonlinear time-series task. Our core contributions are: (i) establishing a unified Volterra-reservoir modeling paradigm; (ii) enabling efficient recursive implementation of infinite-dimensional kernels; and (iii) providing a new framework for dynamical system learning that jointly ensures expressive power and computational feasibility.

To address the high computational cost and difficulty in balancing accuracy and efficiency arising from direct discretization of neural integral equations, this paper introduces spectral methods into the neural operator learning framework for the first time, parameterizing and learning integral operators in the frequency domain. The proposed paradigm ensures both theoretical rigor and computational efficiency: theoretically, it establishes rigorous guarantees on operator approximation capacity and numerical convergence under spectral approximation, grounded in integral equation theory and Fourier analysis; practically, it employs an optimization-driven strategy for solving second-kind integral equations, substantially reducing computational complexity while enhancing generalization and interpolation accuracy. Numerical experiments across diverse nonlinear integral equation tasks demonstrate the method’s effectiveness, stability, and superior efficiency–accuracy trade-off compared to state-of-the-art baselines.

This paper investigates the well-posedness of solutions to the Guyon–Lekeufack path-dependent volatility model under general kernel functions. The underlying dynamics are formulated as stochastic Volterra equations with non-convolutional, unbounded kernels and non-Lipschitz coefficients—a setting that poses significant challenges for modeling trend and activity features. Methodologically, we integrate stochastic Volterra equation theory, path-dependent stochastic analysis, and kernel regularity analysis, complemented by numerical calibration and sensitivity validation. Our key contributions are threefold: (i) we establish, for the first time, global existence and uniqueness of solutions under non-convex, unbounded kernels and non-Lipschitz conditions; (ii) we introduce a novel positivity criterion based on logarithmic derivative inequalities, rigorously ensuring strict positivity of the volatility process; and (iii) we empirically verify that the calibrated kernel satisfies the proposed positivity condition, while demonstrating that exponential kernel choices exert negligible impact on calibration accuracy.

This work addresses the failure of the classical Bismut–Elworthy–Li (BEL) formula for stochastic Volterra processes due to their path-dependent nature. To overcome this, the authors develop a novel integration-by-parts (IBP) formula based on Riemann–Liouville fractional derivatives, which interpolates between the standard chain rule and the BEL formula. Their analysis reveals a counterintuitive smoothing effect: the rougher the noise—characterized by a Hurst parameter $H \in (0,1/2)$—the smoother the resulting expectation functional. Specifically, directional differentiability along constant directions is guaranteed whenever the test function’s Hölder exponent satisfies $\beta > 2H$. The framework is further extended to establish first- and second-order BEL formulas for square-integrable directions in additive noise settings, with applications to forward and rough volatility models that clarify the trade-off between the regularity of the test function and the existence of directional derivatives.

This work addresses the high computational complexity of Volterra signatures with matrix-valued kernels in time series modeling by proposing three efficient algorithms: a general O(J²) approximation, an FFT-based O(J log J) acceleration, and an exact recursive O(JR²) method tailored for state-space kernels. The authors demonstrate that multi-factor kernels do not increase asymptotic complexity. By integrating Chen-type convolution factorization, fast Fourier transforms, state-space representations, and JAX-based automatic differentiation, the proposed methods enable efficient computation of Volterra signatures. All algorithms are implemented in the open-source JAX library “tensordev,” significantly reducing computational costs while preserving the standard complexity dependence on path dimension and truncation order.

This study investigates the boundary behavior of Volterra square-root processes on the non-negative real line, with a focus on whether the process hits zero and the associated probabilistic properties. By establishing a time-dependent Feller condition and leveraging comparison principles for Volterra integral equations, generalized Riemann–Liouville fractional differential equations, and asymptotic analysis via Laplace transforms, the work systematically characterizes boundary behavior and moment properties under different kernel functions. The main contributions are as follows: for regular kernels, the process is shown to stay strictly positive over finite time horizons and to possess finite negative-order moments; for rough kernels, the process hits zero with positive probability, inducing an atom at the boundary, yet its limiting distribution retains finite negative exponential moments. These results provide rigorous criteria for the existence of equivalent martingale measures in Volterra Heston models.

This study addresses the ill-conditioning issue in finite-memory Volterra models under non-Gaussian inputs, which arises from monomial parameterization. To mitigate this, the authors propose performing input-distribution-matched Gram–Schmidt orthogonalization of monomials in the \(L^2(P)\) space, thereby constructing a Volterra–Wiener–Kunchenko (VWK) basis. This approach interprets the VWK basis as a generalized polynomial chaos coordinate system for data-driven Volterra system identification. The work establishes, for the first time, a second-order misspecification penalty theorem and formally verifies—using Lean 4—the Krawtchouk polynomial structure under arbitrary-order binomial distributions. Experimental results demonstrate that the VWK basis substantially improves the condition number of the basis matrix, with the excess risk governed by the input skewness coefficient \(\delta\); notably, this risk vanishes exactly for symmetric inputs and outperforms conventional power bases under finite-sample regimes.

This study addresses the challenges of semiparametric modeling for parameter estimation and hypothesis testing in non-Gaussian stochastic processes. It establishes a unified moment-cumulant framework that, for the first time, formally links finite Volterra models with generalized Kunchenko stochastic polynomials, thereby clarifying the fundamental distinction between minimum mean square error (MMSE)/L2 criteria and Polynomial Maximization Methods (PMM). The work proposes a verifiable research program bridging statistics and signal processing, explicitly identifying three necessary conditions for PMM to achieve efficiency gains in radio engineering applications. An open-source R package, EstemPMM, has been developed to support practical implementation. The project has produced fifteen thesis contributions, significantly advancing both the theoretical foundations of PMM and its real-world application in non-Gaussian estimation problems.